Cheat SheetFinal ExamDifferentiation rules: (f + g)0= f0+ g0(f − g)0= f0− g0(cf)0= cf0(fg)0= f0g + fg0(fg)0=f0g−fg0g2(f(g(x)))0= f0(g(x))g0(x)The most common derivatives:(c)0= 0 (xn)0= nxn−1(ex)0= ex(ax)0= axln a(ln x)0=1x(logax)0=1x ln a(sin x)0= cos x (cos x)0= − sin x(tan x)0= sec2(x) (sec x)0= sec x tan x (csc x)0= − csc x cot x(cot x)0= − csc2(x) (arcsin x)0=1√1−x2(arccos x)0= −1√1−x2(arctan x)0=11+x2(sec−1x)0=1x√x2−1(cot−1x)0= −11+x2(csc−1x)0= −1x√x2−1Two important limits:limx→0sin xx= 1 limx→01−cos xx= 0Implicit Differentiation: When the relation between the dependent variable y and theindependent variable x is given by an equation, we can find y0by taking derivatives of bothsides of the equation with respect to x, treating y as a function of x. In other words, yshould be treated as the variable u in the u-substitution scheme.Related Rates: Sometimes we wish to find the rate of change in time of a certain quan-tity, say A, but we have no way of doing it directly. Usually, we can find the rate of a changeof a related quantity B. In order to find the rate of change of A, it is necessary first tofind an equation relating A and B, and then proceed to differentiate said equation with re-spect to time, treating both A and B as functions of time (like in the u-substitution scheme).Logarithmic Differentiation: In order to find the derivative of a function of the formy = (f (x))g(x), take natural logarithms in both sides of the previous equation and then pro-ceed to differentiate using implicit differentiation.Rolle’s Theorem: If f is a continuous function in [a, b], differentiable in (a, b) such thatf(a) = f (b), then there exists a number c in (a, b) such that f0(c) = 0.Mean Value Theorem: If f is a continuous function in [a, b] and differentiable in (a, b),then there exists a number c in (a, b) such that f0(c) =f(b)−f (a)b−a.1Relations between a function and its derivatives: A function f is increasing at eachpoint x where f0(x) > 0. On the other hand, it is decreasing at each point x where f0(x) < 0.It is concave up at x if f00(x) > 0 and concave down at x if f00(x) < 0.Important Definitions: Let f be a function and c a point of its domain D:1) We say that c is an absolute maximum of f if f (x) ≤ f(c) for all x in D. On the otherhand, we say that c is an absolute minimum of f if f(x) ≥ f (c) for all x in D.2)We say that c is a local maximum of f if f(x) ≤ f (c) for the points x near c and that cis a local minimum of f if f (x) ≥ f (c) for the points x near c.3)We say that c is a critical point of f if f0(c) = 0 or if f0(c) doesn’t exist.4)We say that c is an inflection point of f if f changes concavity at c.The Closed Interval Method: Given a continuous function f in a closed interval [a, b],we can find its absolute maximum and its absolute minimum value by following these steps:1)Find the critical points of f in [a, b].2)Evaluate f at the critical points.3)Evaluate f at the endpoints of the interval.4)Compare values and decide.First Derivative Test: Let f be a function and let c be a critical point of f . If f0changes sign from negative to positive at c, then c is a local minimum of f. On the otherhand, if f0changes sign from positive to negative at c, then c is a local maximum of f . If f0doesn’t change signs at c, then c isn’t either a local maximum nor a local minimum of f.Second Derivative Test: Let f be a function and let c be a point such that f0(c) = 0.If f00(c) < 0, then c is a local maximum of f. If f00(c) > 0, then c is a local minimum of f.Definitions: Whenever we wish to compute a limit of the form:limx→af(x)g(x)and limx→af(x) = 0, limx→ag(x) = 0, we say that our original limit is an indeterminateform of type00. On the other hand, if limx→af(x) = ±∞ and limx→ag(x) = ±∞, we referto our original limit as an indeterminate form of type∞∞.L’Hospital’s Rule: If the limitlimx→af(x)g(x)is an indeterminate form of type00or of type∞∞, then:limx→af(x)g(x)= limx→af0(x)g0(x)Slant Asymptotes: We say that the line y = mx + b is a slant asymptote of the functionf(x) iflimx→±∞[f(x) − (mx + b)] = 0This usually occurs in rational functions whenever the degree of the numerator is 1 higherthan the degree of the denominator.Guidelines for sketching a curve:A. Domain.B. Intercepts.C. Symmetry.D. Asymptotes: Vertical, Horizontal, Slant.E. Intervals of increase and decrease.F. Local maxima and local minima.G. Concavity and points of inflection.H. Sketch the curve.Good luck on the